where \([\underline{x}, \overline{x}]\)  represent variable bounds, \(G_i\)  is the set of generators connected to bus \(i\) ,
and \(c_i^\mathrm {re}\)  and \(c_i^\mathrm {im}\)  are smooth nonconvex functions of \(V_t\)  and \(\vartheta _t\) .
We denote by \(Y = Y^\mathrm {re} + \sqrt{-1} Y^\mathrm {im}\)  the \(|B| \times |B|\)  admittance matrix (with \(Y_{ii} = y_{ii} - \sum _{j \ne i} Y_{ij}\) , \(y_{ii} = y^\mathrm {re}_{ii} + \sqrt{-1} y^\mathrm {im}_{ii}\) ,
where \(y^\mathrm {re}_{ii}\)  and \(y^\mathrm {im}_{ii}\)  are the shunt conductance and susceptance at bus \(i \in B\) , respectively).
Using the polar formulation, the functions \(c_i^{\mathrm {re},\mathrm {im}}\)  can be written, for all buses \(i\in B\) , as [1]}

\(\begin{aligned}c_i^\mathrm {re}(V, \vartheta ) &= Y^\mathrm {re}_{ii} V_i^2 \\&+\sum _{j \in N_i}V_i V_j \big [Y^\mathrm {re}_{ij}\cos (\vartheta _{i} - \vartheta _{j}) + Y^\mathrm {im}_{ij}\sin (\vartheta _{i} - \vartheta _{j}) \big ], \\c_i^\mathrm {im}(V, \vartheta ) &= -Y^\mathrm {im}_{ii} V_i^2 \\&+\sum _{j \in N_i}V_i V_j \big [Y^\mathrm {re}_{ij}\sin (\vartheta _{i} - \vartheta _{j})- Y^\mathrm {im}_{ij}\cos (\vartheta _{i} - \vartheta _{j}) \big ],\end{aligned}\) 
